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Absolutely Convergent, Conditionally Convergent, or Divergent?

  1. Dec 3, 2016 #1
    1. The problem statement, all variables and given/known data

    Σ (-1)n-1 n/n2 +4
    2. Relevant equations
    lim |an+1/an| = L
    lim bn = 0
    3. The attempt at a solution
    So I tried multiple things while attempting this solution and got inconsistent answers so I am thoroughly confused. My work is on the attached photo.
    I found that the ratio test did not work because the limit equaled 1.
    Then I used the comparison test and found that the series was convergent...which I'm really confused about because one can compare n/n2 +4 to n/n2 which is 1/n and is divergent.
    Then I compared values of the function to its absolute function and the values were the same...but I'm not sure what that means...
    I'm also really not sure what the difference is between being absolutely convergent vs conditionally convergent.
    Any help is greatly appreciated.

    Attached Files:

  2. jcsd
  3. Dec 3, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    If you mean what you wrote, the series is obviously divergent, since the nth term is
    $$t_n = (-1)^{n-1}\frac{n}{n^2} + 4$$
    (according to you) and ##\sum_{n=1}^{\infty} 4## is divergent.
    However, if you meant to write
    $$t_n = (-1)^{n-1} \frac{n}{n^2+4},$$
    then that is a different matter entirely.

    Use parentheses to clarify the meaning: a/b+c means ##\frac{a}{b} + c##, while a/(b+c) means ##\frac{a}{b+c}##.
  4. Dec 3, 2016 #3
    I mean tn=(-1)n-1 * n/(n2 +4)
    I am new to the fourm and am still learning how to format things.
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